On the Formal Consistency of Theory and Experiment, with Applications

On the Formal Consistency of Theory and Experiment, with Applications to Problems in the Initial-Value Formulation of the Partial-Differential Equations of Mathematical Physicsy DRAFT Erik Curielz June 9, 2011 yThis paper is a corrected and clarified version of Curiel (2005), which itself is in the main a distillation of the much lengthier, more detailed and more technical Curiel (2004). I make frequent reference to that paper throughout this one, primarily for the statement, elaboration and proof of technical results omitted here. zI would like to thank someone for help with this paper, but I can't. 1 Theory and Experiment Abstract The dispute over the viability of various theories of relativistic, dissipative fluids is an- alyzed. The focus of the dispute is identified as the question of determining what it means for a theory to be applicable to a given type of physical system under given conditions. The idea of a physical theory's regime of propriety is introduced, in an attempt to clarify the issue, along with the construction of a formal model trying to make the idea precise. This construction involves a novel generalization of the idea of a field on spacetime, as well as a novel method of approximating the solutions to partial-differential equations on relativistic spacetimes in a way that tries to account for the peculiar needs of the interface between the exact structures of mathematical physics and the inexact data of experimental physics in a relativistically invariant way. It is argued, on the ba- sis of these constructions, that the idea of a regime of propriety plays a central role in attempts to understand the semantical relations between theoretical and experimental knowledge of the physical world in general, and in particular in attempts to explain what it may mean to claim that a physical theory models or represents a kind of physical system. This discussion necessitates an examination of the initial-value formulation of the partial-differential equations of mathematical physics, which suggests a natural set of conditions|by no means meant to be canonical or exhaustive|one may require a mathematical structure, in conjunction with a set of physical postulates, satisfy in order to count as a physical theory. Based on the novel approximating methods developed for solving partial-differential equations on a relativistic spacetime by finite-difference methods, a technical result concerning a peculiar form of theoretical under-determination is proved, along with a technical result purporting to demonstrate a necessary condition for the self-consistency of a physical theory. 2 Theory and Experiment Contents 1 Introduction 4 2 Relativistic Formulations of the Navier-Stokes Equations 10 2.1 The Three Forms of Partial-differential Equation . 10 2.2 Parabolic Theories and Their Problems . 10 2.3 Hyperbolic Theories . 11 2.4 The Breakdown of Partial-Differential Equations as Models in Physics . 12 3 The Kinematical Regime of a Physical Theory 15 3.1 Kinematics and Dynamics . 17 3.2 Constraints on the Measure of Spatiotemporal Intervals . 19 3.3 Infimal Decoupages . 23 3.4 The Kinematical Regime . 25 4 Physical Fields 30 4.1 Algebraic Operations on the Values of Quantities Treated by a Physical Theory . 31 4.2 Inexact Scalars . 37 4.3 Algebraic Operations on Inexact Scalars . 41 4.4 Inexact Scalar Fields and Their Derivations . 51 4.5 Inexact Tensorial Fields and Their Derivations . 58 4.6 Inexactly Linear Operators . 65 4.7 Integrals and Topologies . 67 4.8 Motleys . 71 5 Physical Theories 73 5.1 Exact Theories with Regimes and Inexact, Mottled, Kinematically Constrained Theories 75 5.2 Idealization and Approximation . 83 5.3 An Inexact, Well Set Initial-Value Formulation . 92 5.4 A Physically Well Set Initial-Value Formulation . 97 5.5 Maxwell-Boltzmann Theories . 101 5.6 The Consistency of Theory and Experiment . 105 6 The Soundness of Physical Theory 106 6.1 The Comparison of Predicted and Observed Values . 106 6.2 Consistent Maxwell-Boltzmann Theories . 110 6.3 The Dynamical Soundness of a Physical Theory . 112 6.4 Theoretical Under-Determination . 114 7 The Theory Is and Is Not the Equations 116 3 Theory and Experiment It is a capital mistake to theorize before one has data. Sir Arthur Conan Doyle \A Scandal in Bohemia" In theory, there's no difference between theory and practice. In practice, there is. Yogi Berra Theory without experiment is philosophy. Allison Myers 1 Introduction In this paper, I intend to investigate a series of questions on the complex interplay between the theoretician and the experimentalist required for a mathematical theory to find application in modeling actual experiments and, in turn, for the results of those experiments to have bearing on the shaping and substantiation of a theory. On the one hand, we have the rigorous, exact and often beautiful mathematical structures of theoretical physics for the schematic representation of the possible states and courses of dynamical evolution of physical systems.1 On the other hand, we have the intuitive, inexact and often profoundly insightful design and manipulation of experimental apparatus in the gathering of empirical data, in conjunction with the initial imposition of a classificatory structure on the mass of otherwise disaggregated and undifferentiated raw data gathered. Somewhere in between these extremes lie the mutual application to and qualification of each by the other. It is one of the games of the experimentalist to decide what theory to play with, indeed, what parts of what theory to play with, in planning experiments and designing instruments for them and modeling any particular experimental or observational arrangements, in light of, inter alia, the conditions under which the experiment will be performed or the observation made, the degree of accuracy expected or desired of the measurements, etc., and then to infer in some way or other from the exact, rigorous structure of that theory, as provided by the theoretician, models of actual experiments so that he may explicate the properties of types of physical systems, produce predictions about the behavior of those types of systems in particular cirumstances, and judge whether or not these predictions, based on the schematic models contructed in the framework of the theory, conform to the inaccurately determined data he gathers from those experiments. It is one of the games 1I follow the discussion of Stein (1994) here in my intended use of the term schematic to describe the way experiments are modeled in physics. That paper served as much of the inspiration for the questions I address in this paper, as well as for many of the ways I attempt to address the questions. Besides to that paper, I owe explicit debts of gratitude for inspiration to Geroch (2001), Stein (npub, 1972, 2004), with all of which, I hope, this paper has affinities, in both method and conclusions. 4 Theory and Experiment of the theoretician to abduct exact, rigorous theories from the inaccurately determined, loosely organized mass of data provided by the experimentalist, and then to articulate the rules of play for those theories, by, inter alia, articulating the expected kinds and strengths of couplings the quantities of the theory manifest and the conditions under which they are manifested, leaving it to the experimentalist to design in light of this information probes of a sort appropriate to these couplings as manifested under the particular conditions of experiments. Jointly, the two try to find, in the physical world, common ground on which their games may be played. No matter what one thinks of the status of these sorts of decisions and articulations in science|whether one thinks they can ultimately be explained and justified in the terms of a rational scientific methodology or whether one thinks they are, in the end, immune to rational analysis and form the incorrigibly asystematic bed-rock of science, as it were|it behooves us, at the least, to get clearer on what is being decided and articulated, and on how those decisions and articulations bear on each other, if, indeed, they do at all . I will not examine the actual play of any current or historical theoreticians and experimentalists in their attempts to find common, mutually fruitful ground on which to engage each other. I leave those issues, fascinating as they are, to other, more competent hands. Neither will I examine all the different sorts of games in which they engage in their respective practices, rather treating only those played in one small part of their common playground, that having to do with the comparison of predicted and observed values of a system as it dynamically evolves for the purposes of testing and substantiating a theory on the one hand, and refining experimental methods and design on the other. [*** For this latter, cf. the suggestion by Lee and Yang of the experiments that showed violation of party; differentiate these more explicitly from the construction of theoretical models that only use well-founded theory to predict, with no thought of substantiation, such as planning the moon-shots ***]. I do not deal explicitly with others, such as predictions that have nothing to do with comparison to observations (for instance, the use of Newtonian gravity in calculating trajectories during the Apollo project's flights to the Moon), or the calculation of fundamental properties of physical systems based on theoretical models (for instance, the use of the quantum theory of solids to calculate the specific heat of a substance). [*** Distinguish \comparison to observation" from \use of observation" in these examples|for in the moon-shots they surely also compared the observed results of previous moon-shots to, among other things, refine their methods of prediction and characterization for future ones ***].

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